A Survey of Numerical Methods for Parabolic Differential Equations

Abstract

The chapter introduces the theoretical view of finite difference methods for approximating the solutions of partial differential equations of parabolic type. A few preliminary definitions and facts about difference analogues of derivatives are first presented. The symbols “u” and “w” will be used to denote the solution of a differential equation and the solution of a difference equation, respectively. Numerical treatment of parabolic differential equations is done by considering the boundary value problem for the heat equation in one space variable. The chapter begins by deriving the backward difference equation and the Crank-Nicolson difference equation. The local error in the time direction is decreased by deriving the Crank-Nicolson difference equation. Crank-Nicolson equations can be applied to problems for which slope conditions are specified at a boundary, but the disadvantage of the Crank-Nicolson method is that greater smoothness is required of the solution of the differential equation to insure convergence. The chapter deals with unconditionally unstable difference equations and higher order correct difference equations, and presents a comparison of the calculation requirements.